Triangulation of semi-analytic sets

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

S. L OJASIEWICZ

Triangulation of semi-analytic sets

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3

^e

série, tome 18, n

^o

4 (1964), p. 449-474

<http://www.numdam.org/item?id=ASNSP_1964_3_18_4_449_0>

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by

^S. LOJASIEWICZ

(Krak6w)

The

triangulability question

^for

algebraic

^{sets was first} considered

by

van de Waerden

[15]

ⁱⁿ

1929,

^{and for}

analytic

^sets

by

^Lefschetz

[5], ^Koop-

man and Brown

[4],

and Lefschetz and Whitehead

[6],

in 1930-1933.

In fact, j5]

^and

[6]

deal with

triangulation

^{of «}

analytical complex

^»

(which

^{is a fi-}

nite

disjoint

collection with

compact

^{union of subsets of}

1R~

each

being

^an

open and

relatively compact

^{subset of an}

analytic

^{subset of an} open set in A lack of ^a convenient

technique

^{at that time was}

probably

^{the reason}

for the

proofs being

rather sketched.

Therefore, according

^{to an}

opinion

^of

many

mathematiciens,

it is of some interest to

give

^{a new detailed}

proof.

This is

just

the purpose of the

present

paper, and

actually

^we

give

^{a so-}

mewhat more

general

result: simultaneous

triangulation

^{of any}

locally

^finite

collection of

semi analytic

^{subsets of}

^1Rn

or,

owing

to the Grauert

imbedding

theorem

[3],

^{of any countable real}

^analytic

^manifold.

Moreover,

^{our formula-}

tion of the result is somewhat more

precise;

^it follows e.g. that in any

exemple

^{of Milnor’s}

type [9]

^the

homeomorphism

between the

polyhedra

^{has to}

have a

singularity

^of

non-algebraic type (it

^can not have the

property (A) (see § 3)).

However the frame of idea of the construction we

give

^{is that of}

Lefschetz

([5]

^and

[6]).

The

semi-analytic (resp. semi-algebraic)

^{sets are} those which can be lo-

cally given by analytic (resp. algebraic) inequalities (see [13], [14]

^and

[8]).

We may observe that ^the

body

^{of an «}

analytic complex »

^is

exactly

^the

same as a

compact semi-analytic

set. We will need certain basic

properties

of

semi-aualytic sets ;

^{3 these are}

^only

^stated

^{in §} 1,

and will be contained with

proofs

in papers to appear

separately.

The method we use is that of normal

decompositions

^of

semi-analytic sets ;

^{it follows an} old idea of

Osgood [10] (see

^also

[11]),

^{and was}

applied

Pervenuto alla Redazione il 18

Maggio

^1964.

by

the author in

[7] (1).

The normal

decompositions

^{are certain}

special

^local stratifications in the sens of Thom

[14].

For the semi

algebraic

^{case an ele-}

gant Whitney’s

stratification

[17]

^{can be} also used.

The

result

^was communicated on the

Congress

ⁱⁿ

Stockholm, 1962,

^and

the construction was

presented

^{in details on a} seminar at the Istituto Ma- tematico Leonida Tonelli of the

University

ⁱⁿ

Pisa, spring,

^1963.

Afther

having

written the

manuscript

^{the autor observed that had}

just appeared

^{a thesis of B. Giesecke}

[2] concerning

^{also the}

triangulation

^of

semi-analytic sets;

because of the differences which seem to exist in results and methods

used,

^our article may be however of

independent

^interest.

One should also mention a paper of K. Sato

[19]

^{on local}

triangulation

of real

analytic

^varieties.

§

^1.

Preliminaries

on

semi-analytic ^sets.

In

this §

the results are

only stated;

^the

proofs

will be contained in articles to

be published

^soon.

I.

Definition of

^seini

analytic

^set. Let ll~ be a real

analytic

^manifold.

Let

A, G c M

^{and let} ... ,

fr

be real functions

(each

^{one defined on}

a subset of

ill);

^we say that A is described in Q’

by 11 , ... , fr

^iff

fi ,

^{... ,}

fr

are defined in Q~ is a finite union of finite intersections of sets of the

form x

^{E G :}

f) (:x) &#x3E; 0 )

^or

( x

^{E G :}

Ii (x)

⁼

0 ) ;

^we say that A is des- cribed at c E M

by It , ... , fr

^{iff it} is described

by

these functions in a

neig-

hborhood of c.

,A subset A of M is said to be semi

analytic

^{iff it can} be described at any ^{c E} ill

by

^{a set}

(depending

^on

c)

^{of real}

analytic

^functions

(at c). Equi- valently, A

^is

semi analytic

iff for each x E M the _germ of A at x

belongs

^to

the smallest class S of germs at x

(of

^{subsets of}

satisfying

u, v ^E S &#x3E;

u U vi

u BB v

^E

~’,

^and

containing

the germ ^{at x} of each set of the form

( f &#x3E; 0 )

with

f

^real

analytic

^{in a}

neighborhood

^{of x.}

The union of any

locally

^finite

family

^{and the} intersection of any finite

family

^of

semi-analytic

^{sets is}

semi-analytic.

^The

complement

of any semi-

analytic

^{set is}

semi-analytic.

^A subset of any closed submanifold

(2) Mi

^{of M}

is

semi-analytic

ⁱⁿ

Mi

^{if and}

^only

if it is semi

analytic

^{in The}

image

^of

any

semi-analytic

^set

by

^an

analytic isomorphisme

^is

semi analytic.

^{A finite}

product

^of

semi-analytic

^{sets is}

semi-analytic.

(’I

^{See also}

[8],

where another application of this method is given.

(~)

A submanifold of M is a subset of M which is locally the inverse image by ^a

chart of a linear subvariety (of ^{the same} dimension for _any point ^{of this} set), ^{with the}

induced manifold structure.

If.

decompositions.

^A function B

(Zt ,

^{... ,}

holomorphic

^at

(0, .., , 0 ; 0)

is called ^a

distinguished polynomial

in x iff it is a

polynomial

in z with coefficients

vanishing

^at

(o, ,.. , 0) except

^the

leading

^{one which}

=1;

^{it is said to} be real iff its values for real

arguments

^{are real.}

Let M be ^a real

analytic

manifold of dimension n. Let c E M. A normal

system

^{at c is a}

couple

^{of an}

analytic

chart g :

at c (3)

^such

that

g (c) = 0,

^{and a}

system

^{where 9}

^xl)

^{is a real}

distinguished polynomial in Zz

with discriminant

~..., xk)~0,

such that in some

neighborhood

^of

(0,

^{... ,}

0)

A

neighborhood Q

⁼

g-1 (Qo)

of c, where

90 = x : ~ I Xi I C

_{C 9}

(C~)~

is said to be normal

(with respect

to the above normal

system)

^iff

Hlk

^are

holomorphic

^on

satisfy (a)

^and

(fl)

ⁱⁿ

for with

0 ~ ~

^n. Thus

~1o

^{is a normal}

neighborhood

^{of 0}

following

^{the normal}

system (e,

^{e :}

^Rn

^-+

^JR" being

^the

identity

map.

Every neighborhood

^{of c contains a normal one.}

The normal

decomposition of Q (following

^the above normal

system)

^is

the

decomposition

--

where

rk

^are the connected

components

^of

(we put

^for convenience

.go 1=1).

^{The sets}

rxk

^are called members of the

decomposition.

^Thus

1’x

⁼

g-l

^where

7~

^are the members of the nor-

mal

decomposition

^of

Qo ^following

the normal

system (e,

^{at 0.}

A normal

decomposition at

^c is the normal

decomposition

^{of a normal}

neighborhood following

^{a normal}

system

^{at c.}

(3) ^{i. e. local} analytic coordinates system ^{at c.}

1. Any

^normal

decomposition

is finite.

be a normal

decomposition.

union of ^some

be a normal

decomposition

^{at () E}

following

^{a nor-}

mal

system

with S~ _open

(in and analytic

^{in Q such that}

0 E Q,

^{= 0}

in D and lim ,

u-0

4. Any

^member

Tx

^of any normal

decomposition

^{at c is a} k-dimensional

analytic

submanifold

(F,"

^are open and

ro

^_

(0))

such that c E

rx .

6. Let

Q = k, U

^{be a normal}

decomposition

^{as in 3. Let 0 m n.}

Jk, ?

Denote

by a

^the

projection Rn

3

xn) 2013(i?-..? x»,) E 1Rm

^and

bye.

^the

identity

map

1Rm -+ 1Rm.

^The

couple (e., [Hlk

^{a normal}

^system

at 0 E

1Rm

^and

Q*

^{= n}

^(Q)

^{is a normal}

neighborhood ;

^let

Q*

⁼

^U

" be its normal

decomposition.

^{Then for} any

7~

^{with k S m we have a =}

rk

(for

^some

^x’).

III. Existence theorems. A normal

decomposition Q

⁼

k, U

^{A:, x is said to}

be

compatible

^{with a}

function f

^defined

in Q

^iff

f

^{= 0 or}

f #

^{0 on any mem-}

ber

r:;

it is said to be

compatible

^{with a subset A of ~VI iff any member}

is contained in A or in

M’" A.

^If A is described at c

by f1,

... ,

fr ,

^{7 then}

any normal

decomposition

^{of a}

sufnciently

^small

neighborhood

^{at c which}

is

compatible

^{with ... ,}

fr

^{is also}

compatible

^{with A.}

1. Let c E M. There is a normal

decomposition

^{at c which is}

compatible

with

given

^functions

fi 7 ....f, analytic

^at c, resp. with

given semi-analytic

sets

At ,

^{... ,}

AS

^C

1JI;

the normal

neighborhood

^{can be chosen}

arbitrarily

^small.

Let M be an afiline space, let

f

^{be an}

analytic

^{function at c E M. A line} A

though

^{c is} said to be

singular for f’

at c, iff

f

^vanish

identically

^{in a}

neighborhood

^{of c in A. Let A be a}

semi-analytic

^{subset of M. A line A}

through

^{c is said to be non}

singular for

^{A at} c, iff A ^can be described at

c

by

^{a set of functions for which A is}

non-singular

^{at c.}

2. Let _... be

analytic

^{functions at c E}

M,

^{such that}

fj

^{= 0} =&#x3E;

F = 0 in a

neighborhood

^{of c}

( j =1, ... , r),

^{let A be a}

non-singular

^{line for}

... ,

fr,

^and

let x

^{be a}

hyperplane

^{such that 1} n X ⁼

(c).

^{There is a}

normal

decomposition Q

⁼

following

^{a normal}

system (g, ( H/ ))

^{at e}

which is

compatible

^with

fi , ... , fr

^{and such}

that g

^is

affine, g (1)

^{is the}

(i.e.

gj ^{= 0} on I

for j = 1, ... , rc -1), g (X)

^{is the ... ,}

plane (i.e. gn = 0 on X),

^and

F (r) = 0) ^(i.e. H,l’=0 =&#x3E;

^lJ’=O

in

Q).

The normal

neighborhood Q

^{can be chosen}

arbitrarily

^small.

3. Let

A , ... ,

^be

semi-analytic

^{subsets of}

M,

^{let  be a}

non-singular

line for

A~ , ... ~ A$

^{at c E M and}

^{let X}

^{be a}

hyperplane

^{such that}

~, ~ x = ~c).

There is ^a normal

decomposition following

^{a normal}

system (g,

^{at c}

which is

compatible

^with

Ai ~ ... ,

^{and such}

that g

^is

affine, 9 ⁽¹⁾

^{is the}

xn-axis and g (X)

^{is the}

(Xi’.’" xn-l)-hyperplane.

The normal

neighborhood

can be chosen

arbitrarily

^small.

Vf. Some

properties.

^{Let M be a real}

analytic

^manifold.

1. A subset A of M is

semi-analytic

^{if and}

only

if for any ^c E M there is ^a normal

decomposition

^{at c which is}

compatible

^{with A.}

2.

Any

^connected

component

^{of a semi}

analytic

^{set is}

semi-analytic.

3.

Any

^{member of a normal}

decomposition

^is

semi-analytic.

4. The

closure,

the interior and the

boundary

of any

semi-analytic

^set

are

semi -analytic.

V.

Projection

theore~n. Let M be a real

analytic manifold,

^{~1 an affine}

space. A subset A of M

&#x3E; A

is said to be

partially semi-algebraic (with respect

^{to iff} any c E M has a

neighborhood U

such that A can be des- cribed in U X A

by

^{a set of}

analytic

^{functions which are}

polyno -

mials in x ; thus any

partially semi-algebraic

^{set is}

semi-analytic.

^{The inter-}

section of any finite

family

^of

partially semi-algebraic

^{sets is}

partially

^semi-

algebraic.

^{A union of a}

family

^of

partially semi-algebraic

^{sets is}

partially semi-algebraic, provided

^{that each}

point

^{of M has a}

neighborhood

^{U such}

that U x A meet

only finitely

many sets of this

family.

1.

Any

connected.

component

^{of a}

partially semy-algebraic

^set is par-

tially semi-algebraic.

2. SEiDENBERG ^THEOREM

(4).

^Let

Ao be

^another affine space and denote

by

^{11: the}

projection

^{M X A X}

Ao

³

(u, x, y)

^-+

(u, x)

^{E M X A. If a subset A}

(4) Formulated in

[13]

^for

semi-algebraic

^sets.

5. Annali della Scuola Norm. Sup- -

of x

!lo

^is

partially semi-algebraic

^with

respect

^{to d X}

Ao,

^{then n}

(A)

is

partially semi-algebraic

^with

respect

^{to A.}

VI.

Semi-algebraic

^{sets. A} subset A of an affine space M is said to be

semi.algebraic

^{iff it can be described in IVI}

by

^{a set of}

polynomials ;

^{it is}

said to be

locally semi-algebraic

^{iff it can be described} at any ^{c E} M

by

^a

set

(depending

^on

c)

^of

polynomials;

each bounded

locally semi-algebraic

set is

semi-algebraic.

^{Let S~ be an open subset of}

M ;

^an

analytic

^function

f : Q - 1R

^{is said to be}

analytic algebraic

iff there is ^a

polynomial P(t, x)fl0

such that

P (;x, f (x)) = 0

^{in ~. A} subset A of .M is

locally semi-algebraic

if and

only

^{if it can} be described ^at any ^{c E} M

by

^{a set}

(depending

^on

c)

^of

analytic algebraic

^functions

(at ^c).

All the facts stated in

this §

remain valid for M affine

(and

for affine charts

g)

^{if we}

replace

the notions of semi

analytic

^{set and}

analytic

^function

by

^{those of}

locally semi-algebraic

^{set and}

analytic-algebraic

^function.

Let P be the

projective

space derived from ^a finite dimensional vector

space V (identified

^{with the set of all lines}

through

^{0 in}

V) ;

^{the canonical}

map

(of P)

^{is the} For any

projective hyper- plane P’ C P (derived

^{from a}

hyperplane

^{V’ of}

V)

^{the set}

P BB P’

^{with its}

natural affine structure

(such

^{that for}

any v E V B

V’ the restriction nv+ v:

v -E-

^{is an affine}

isomorphism)

is called ^an affine chart of P.

Any

affine space ^can be considered as an affine chart of a

projected

space.

Consider a

multiprojective

space i.e. ^a finite

product

of finite dimen- sional

projective

^vector spaces R ⁼

P,

&#x3E; ^... X

Pk.

^{The canonical} map

(of R) ( ViB 101)

^{X ... X}

^{( Veg} JOJ)

^--~

R,

^where

:f, ^are the canonical _maps of

Pi,

^{y and}

^Vi

^{are the vector spaces from which}

P;

^are

derived ;

^an affine chart of R is a

product ~li

^{x ...}

X Ak where Ai

is an affine chart of

Pi, i =1 ~ ... , lc.

^A subset A of R is said to be semi-

algebraic

^{iff a-’}

(A)

^{is semi}

algebraic.

^{A subset of an} affine chart A of R is semi

algebraic

in R iff it is

semi-algebraic

^{in A. A subset of R is semi-}

algebraic

^{iff it can be described} at any ^{c E} R

by

^{a set}

(depending

^on

c)

^of

polynomials

^{in an} affine chart of R. The union and the intersection of any finite

family

^of

semi-algebraic

^{sets and the}

complement

of any

semi-algebraic

set are

semi-algebraic ;

^{a finite}

product

^of

semi-algebraic

^{sets is}

semi-algebraic.

Let

R1, R.

^be

multiprojective

spaces : ^a map of a subset of

R1

^into

R~

is said to be

semi-algebraic

^{iff’ its}

graph

^is

semi-algebraic.

^The

composition

of

semi-algebraic

^{maps is}

semi-algebraic.

^The

image

and the inverse

image

of any

semi-algebraic

^set

by

any

semi-algebraic

map is

semi-algebraic.

§

^2.

Some

lemmas.

For _any

C1.maiaifold

d denote

by Au

^its

tangent

space at u ^E

A ;

^for

any

A’,

^{where d’} is another

Ci.manifold,

^let

dgu :

be its differential ^at u ;

put ranku g

^{= dim}

dgu (Au)

and rang g = sup

(ranku g :

If

ranku

g ⁼ dim

A’,

then g

(u)

^{is an interior}

point

of g

(.if).

^Hence

if

g (A)

^{has no interior}

points,

^{then rank}

g

^dim

^A’.

LEMMA 1

(Sard (5)).

^Let

A,

^{A’ be real}

analytic

^countable

(6)

^manifolds

and

let g :

^{~1-~ ~.’ be}

analytic.

^If

rank g

^dim

A ’, then g (d)

is meager

(7).

PROOF. The lemma

being

trivial when dim ~1=

0,

^assume it true if dim ~1

n

and let dim ~1= n

&#x3E;

^{0. Let u E}

A ;

^{for some}

neighborhood

^{U of}

u the set Z

= ~u

^E

U; ranku g

^rank

gu)

^is

semi-analytic

and nowhere dense in

t7; let Q

⁼

k, U

^{be a normal}

decomposition

^{at u which is}

compatible

with Z. It is sufficient to prove that

each g (Tx )

^is meager. If k

n,

^{it is}

true

by

^{the induction}

hypothesis.

Consider any Since

ranku g

= p in

Ty

where p ⁼ rank gu dim A’. Therefore any

point

^of

ry

has a

neighborhood

^whose

image by g

is contained in a

p-dimenf3ional

^sub-

manifold of ~1’ and

hence g (I"’)

is meager.

LEMMA 2

(Lefschetz- Whitehead) (8).

^{Let A be a}

01. manifold.

Let M be

an m-dimensional affine

space, 1T

its vector space, ^{and S an}

sional Ci.submanifold of V such that u E 8 &#x3E;

u f Su (9).

Let g : A -+ M and h : ~1--~ S be

Ci-maps.

If the map T: A x

1R

3

(u, l)

- g

(u) + lh (u)

^{E M is}

of rank

m - 1,

then rank h m - 2.

PROOF. Let u E A.

By assumptions

When A

=1= 0,

it follows the same for the and hence

(letting A

^-+

oo)

^{for the}

(5) ^{This is a} special ^{case of} Sard theorem

[12].

(6) ^{i. e.} with countable basis.

(7) ^{i. e. a counable union} of nowhere dense sets.

(8) ^See

[6],

^{pp. 513-514.}

(9) Any tangent space of a submanifold of M or V is identified with a vector sub- space of V.

map

whence dim ⁱ

REMARK. It is sufficient to assume

that (p

is of rank n~ -1 in an

open set

containing

^{.~ X}

10). For, u being fixed,

^the condition : rank

dggu, 1

~ ~n -1 can be

expressed by vanishing

^{of some} determinants which ^are

polynomials

^{in A.}

Let M be ^an affine space, ^{V its vector} space ; denote

by

^{P the}

projec-

tive space derived from V

(the

set of directions in

M).

Let D be an open subset of M and let

f :

^be

analytic.

^{A di-}

rection a E P is said to be

non-singular for f

iff for each ^c E S the line c

+

^a

is

non-singular

^for

f

^{at c.}

LEMMA 3

(Koopinan

^{and Brown}

(1°)).

^Let

f

^be

analytic and fl

^{0 in an}

open connected Then the set of all

singular

directions

(for f)

^is

meager in P.

PROOF. Put m ⁼ dim M. The

ellipsoid

^{S =}

ju

^E

(u) I = 1),

^where

y : V -+

1Rm

is a

(linear) isomorphism,

satisfies the

assumption

^{of the lem-}

ma 2. Let

in a

neighborhood of ~

and let n : Q X V ₃

(u, v)

^{--+ v E V. Since the set in}

question

^{is the}

image

^of

n (0) by

the local

homeomorphism

it is sufficient _{to prove} that n

(0)

is meager ⁱⁿ S. We have

which

implies

^{that 0 is an}

analytic

subset of !J X V

(for

^the

ring

^of germs of real

analytic

^{functions at a}

point

^is

noetherian).

^{Let c E I) X V and let}

Q

^{= be a normal}

decomposition

^{at c which is}

compatible

^with

9 ;

thus

Q n 0

^{is a union of some}

Af

and it is sufficient _{to prove}

is

meager in 8 for any Consider two

analytic

maps g :

A 3

3

(u, ro)

^{- u E M and h =}

A 3 (u, v)

^{- v E}

V,

and then the A X

x1R3(~)~~)+~M~.

^{Since For any}

x E

A, (g (x), h (x))

^{= x E} 0 and hence

f (g (x~) (x))

^{- 0 in a}

neighborhood

(to)

^See

[4],

p. ^242.

of 1 ⁼ 0. Therefore

1))

^{= 0} in the set

which is open in tl X

1R

and contains A x

(0).

^This

implies that rr (1’)

has no interior

points (in M),

whence the rank

of 97

ⁱⁿ

^A*

^is

S m -1.

By

the lemma 2 and the

remark,

^{rank h m - 2 and}

hence, by

^{the lemma}

1, 3t (A)

⁼

h (A)

^is meager

in S, ^Q.

^{E. D.}

Let A be a

semi analytic

^subset of M. A direction a E P is said to be

non-singular for

A iff for each c E 1~ the line c

+

^{a is}

non-singular

^{for A at c.}

LEMMA 4. Set

(Bv)

^{be a} coutable collection of

semi-analytic

^{sets. The}

set of all directions which are

simultaneously non-singular

^{for each}

^B,

^is

dense in P.

In

fact,

consider any

Bv ;

^{there is a countable}

covering

^{of M}

by

open connected

Wi

such that

By

^can be described in

Wi by

^a finite set of func- tions

analytic and @

^{0 in}

Wz;

since every direction which is simulta-

neously non singular

^{for all}

lij

^{is also}

non-singular

^for

By ,

it follows from the lemma 3 that the set of all

singular

directions for

By

is meager.

Consider now the affine space M ^X

1R

and

put n

^{= dim}

(M

^X

1R), (i.

^e.

dimM=n-1).

Using

^the Weierstrass

preparation

^{theorem we derive}

easily

the follo-

wing

^lemma.

LEMMA 5.

Any semi-analytic

bounded subset of for which the direction

(0) ^{x 1R} (where

^{0 is the zero of}

V)

^is

non-singular

^is

partially

semi

algebraic (with respect

^to

1R).

Denote

by n

the map M ^X

1R 3 (u, ^t)

^{-+ u E M. An}

analytic

submanifold 1P ^C 111 _X

1R

^is said to be

topographic (11)

^{iff 3t}

(y)

^{is an}

analytic

submanifold of M and n1p: y ^{--+ 3t}

(y)

^{is an}

analytic isomorphism : then tp

^{is the}

graph

of an

analytic

function a

(y)

^--~

^1R

^{which we}

identify

with y. The

following

lemma is trivial.

LEMMA 6.

Let V -

^{be an}

analytic

submanifold of Introduce a

euclidean norm in M and let A

&#x3E;

^{0. If}

then y

^is

topographic (and

ⁱⁿ

n(y)).

^If

(ii) This convenient

terminology

^was

proposed

by A. ANDREOTTI.

then the

image of y by

^the map

is also

topographic.

We say that ^a subset Z of has

property (P )

^{iff the} map is open.

Any topographic

submanifold

(of

^M &#x3E;

1R)

^{of dimension}

n -1 has

property (P).

LEMMA 7. For any function

f analytic

^at

(c, 0)

^{E M} &#x3E;

1R

^{and such that} there exist a

neigborhood D’

^{and a}

function g

such that

f,

g

are

analytic

ⁱⁿ

U, g ^{(c, t)} =1=

⁰ and the set

(x

^{E U :}

f (x) g (x)

⁼

0)

^{has pro-}

perty (P).

PROOF. Let H

(u, t)

^{be a}

distinguished polynomial

^{in t at}

(c, 0) (12)

^such

that

&#x3E; f = 0

^{in a}

neighborhood

^of

(e,O),

its discriminant

D (u) Q

⁰

(13).

Let A be a

non singular

^{line for D at} c ; ^{we can}

identify

^{l~ with}

M i &#x3E; 1R (where M1

^{is an} affine space of dimension m -

2)

^{in such a} way that

c ⁼

(c~ 0)

^{and I =}

(cj X ^1R

^{for some c, E}

M1;

^{thus D}

(e1 , s) ~

^{0. Let}

Hi (v, s)

be a

distinguished polynomial

^{in s at c such that 0} -&#x3E; D ⁼ 0 in

a

neighborhood

^{of c.} Consider the

analytic

^function

F (v, t)

defined in ^a

neighborhood

^of

(e, ) ^{0) by}

the formula

where ~j

^{are the roots of}

H1 at v (14) ;

^{we have then}

F (C1 ,

^{0 and}

D

(v, s)

⁼

H (v, s, t)

^{= 0} =&#x3E; F

(v, t)

^{= 0 in a}

neighborhood

^of

(Vi’ 0, 0).

^As-

sume n

= 2,

^or

n &#x3E;

² and the lemma true for n - 1. There exist an open

neighborhood U,

^of

(c, , ⁰⁾

^{and a}

function g

^{such that}

F, g

^are

analytic

ⁱⁿ

Ul, g (ci , t) ~ 0, F = 0 =&#x3E; g = 0

ⁱⁿ

U1,

^{and the set}

t)

^E

Ut : 9 ^{(v, t)}

⁼

Oj

has

property (P ), (in Mi

&#x3E;

1R).

^In

fact,

^{if n = 2 we}

put g

⁼

F;

in the second

case we take

Ui

^{and G for F}

according

^to the lemma

(assumed

^{true for}

n -1)

^{and we}

put g

^{= FG. Now we choose an} open

neighborhood U

^of

(c! , 0, 0)

^so that all the above relations hold and

(v, t)

^E

UI,

^if

(v,

s,

t)

^{E U.}

(12) ^{i. e.} analytic ^at (ol 0), polynomial ^{in t} with coefficients vanishing ^{at c} except the leading ^{one which -1.}

(13)

^{For the existence see e.} g.

[7],

^{nO 11.}

(14) ^{See e.} g.

[7],

^nO 13; ^we put ^{F =1 when} Hi ^{is of} degree ^0.

Then we have

Since both sets on the

right

^{side has}

property (P) (the

^{first one is}

locally

a

topographic

submanifold of dimension

n - 1),

^so has the set on the left.

Therefore the lemma is

proved by

^induction.

By

^{lemma 7}

and § 11 Ill,

^{2 we obtain}

LEMMA 8. Let

Bi ; ... , ^Br

^be

semi-analytic

^{subsets of M x}

1a

for which the line X

’I~

^is

non-singular

^at

(e, y)

^{E M X}

^JR.

There exists a normal

decomposition Q

⁼

k, U x rk ^following

^{a normal}

^{system (g,}

^at

^{(0, y)}

^which

is

compatible

^with

B~ , ... , Br

and such that g is

affine,

g

((c~

&#x3E;

1R)

^{is the}

xn-axis,

g

(M

^X

(y)

^{is the}

xn-l)-byperplane

^and

Vn-I U

^... U

Y° _

=

k U

^has

^{property (P ).}

The normal

neighborhood

^{can be} chosen arbi- k n

trarily

^small.

From § 1, II, 3,

⁵

and V,

^I

we get :

LEMMA 9. Let

Q

⁼

^U T#

^{be a normal}

decomposition following

^{a normal}

k, ^x

system (g,

^at

(e, y)

with g affine and such that g ^X

lR)

^{is the xn-}

axis _{and g}

(M

^a

(y))

^{is the}

(Xt,..., xn-1)-hyperplane.

^{Then all}

rk

^are par-

tially semi-algebraic

and-those with k n are-

topographic.

^{There is a normal}

decomposition Q*

^{= n}

(Q) ^{= U} ,1~’,~~

such that for any

F., k

^{with k n we}

k k

&#x26;

have a

(ruk) ^=1’~x-

^{for some MI.}

We call

Qn

⁼

^U

^the

projected decomposition (of

^the

previous one).

J, ^t

LEMMA 10. Let ~y be defined and

analytic

ⁱⁿ

(open) neighborhoods

of c E

M,

^{v =}

+ 1,

^-~-

2,

^{... ; assume that the} sequence g2+

(e)

^is

strictly

^in-

creasing,

lim 99,

(0)

⁼ 00, lim _99,

(0)

= - 00, and that is

locally

^finite

v -- 00 V - - 00

(as

^a

family

^of

graphs).

^{Let (~ be a} subset of M whose interior contains

(c~

^X

^1R.

There exist restrictions of _{~y to} open connected

neighborhoods

of c and an

analytic isomorphism

g : M X

It

^{--~ M X}

1R

such that : 2°, _{tp" = 9}

(q;:)

^are

disjoint, bounded, semi.analytic, topographic.

3°. Any

bounded subset of M is contained in

some n ~~y : ~ I " [ h N~,

where

Qv

^{= n}

uniformly

in any bounded subset of LU.

with an A inde-

pendent of v,

^{for a euclidean norm in V.}

PROOF. Introduce a euclidean norm in V and

identify

^{M with V so}

that c ⁼ 0. We may ^assume

(without

^{loss in}

generality)

^that

(0) - 1

and ~1

(0) &#x3E;

^1.

Choose a, &#x3E;

^{0 in such a} way that _g2v is defined in

Cy

^where

Uv

⁼

(u ^I ay~,

^{and g~y}

(!7y) c

^where

^4v

^are

disjoint compact

^in-

tervals,

^{contained in}

I &#x3E; 1). Put 0/:

^{= Take a function} y ^on

1R

which is

positive

and constant on each

A,

^{as well on each}

component

^interval

of

A,,

^y and such that _y

(t)

^{acy on}

Av

^and

l(u, t) :

^y

(t))

^c

c G

B ~ By

^the

Whitney approximation

^theorem

(15)

^{we can find}

a function h

positive

^and

analytic

^on

^1R

^{and such that:}

where ky

^{is a constant}

satisfying ) I

for u,

u’ E

U, .

The inverse

fl : ^1R

^-~

^1R

^{of t -+}

(et

^{- -}

e-t)

^{is an}

increasing analytic isomorphism satisfying P (ip 1)

⁼

ip 1 and I I ^min (2, llt)

in

1R.

We will prove

that 9

⁼ g2 ^o g1 ~ 1 where

is ^8.n

analytic isomorphism

^which

satisfies, together

^{with the conditions} 10.6~.

First observe that

((u, t) : ~ t ~ ~ I )

^{= 9}

(Z),

^where

which

yields

^I

we

get ^60. Now,

^{we have}

where b,

⁼

inf (

^:

6 Jy), since,

^{because (}

have This

gives

³⁰

(for b."

^{-~ o0}

as I

^-

oo), and,

^after

showing

^that yy ^are

topographic,

^the

property

40 will be ^a conseqnence of

lim

^y

⁽⁰⁾

⁼

^ip

^oo and the fact that the col- lection

(~~)

and hence the collection

(yv)

^is

^locally

finite in Since

are

analytic

submanifolds which are

disjoint,

bounded and

semi-analytic,

so are

Thus,

ⁱⁿ view of the lemma

6,

it remains to

verify

^that

with an A

independent

^{of v,}

Now, putting Xv

= g,

(Qv‘),

^{we have}

(v, t),

- - ... - . - . - . - . , -

Q.

^{E. D.}

For any . ^:. E -

’R

^such

that q; ^(u) q’ (u)

^{in E}

put

thus in

particular [ffJ, 99]

^{= 99 ; if moreover in}

E,

^we

put

The

following

^{lemma is trivial.}

LEMMA 11. If _q;

(u) q;’ (u)

in L and the closures ¡¡;,

(pl

^{are bounded}

functions I

Let F be a subset of another affine space. N and

satisfy

1jJ

(v)

~

y’ (v)

in 1~: Assume that ^a map g : ~ -~ F satisfies the

following

^condition

By

the map associated

with g

^{we will mean} the map defined

by

^the

following

^formulas

if _g

gJ’

^{in E}

and y y’

ⁱⁿ

F, by

^the

map ho : (99, g’)

--&#x3E;

1p’)

^associa-

ted

with g

^{we will mean the map defined}

by

the first formula. The

following

two lemmas are obvious.

LEMMA 12. We have

hl , ^{h’ C h}

for the maps ^-~ 1p,

hi : g~’

^--~

1p’

associated with g,

and, if 99 m’

ⁱⁿ

E, y y’

ⁱⁿ

F, then ho ^{C la}

^{for the}

map ho : (gJ, cp’)

^-~

jy, y’)

associated with g. If A

c: E and g (A)

^{c B C}

F,

then we have for the map h* :

[99A ,

^--~

[y~B , 1J’B]

associated with gg: A--~B.

LEMMA 13. If

,E,

^{F are}

analytic submanifolds, (p, V analytic,

^and

g : E - F is an

analytic isomorphism,

^{then the} associated ₉₉ --+ 1p is also an

analytic isomorphism.

^{If moreover}

99’, 1p’

^are

analytic,

99

g~’

1jJ

y’ on F,

^then

99~), (W, y)

^are

analytic

submanifolds and the associated map

((p,.p’)

^-+

VI)

^{is also an}

analytic isomorphism.

We say that ^a map

f (of

^{a subset of an} affine space

M ’)

^{into an affine}

space N’ has

property (A-1),

^{iff its}

graph

^is

partially semi algebraic

^with

respect

^{to N’.}

LEMMA 14. Assume the condition

(s)

satisfied. If E is

compact,

q,

99’,

^y,

y’

continuous, and g continuous,

then the associated

map h : [g~,

^-~

[y, y’]

is also continuous.

If g

^has

property (A-1),

g,

99’

^are

partially semi-algebraic

with

respect

^to

1R, and 11’. y’ semi-algebraic,

then h has also

property (A-i).

PROOF. In view of the condition

(8),

^the

graph

of h is the set

Under the

assumptions

of the first

part

^{of the lemma} this set is

compact

and hence the conclusion follows. Under the

assumptions

^{of the}

second,

this set is the

projection by

the map

(u, t, V7 8.,27y r~’, ~, ~’)

^--~

(u, t,

v,

8)

^{of the}

intersection of the

following eight

subsets of the

(u, t,

v, 8, q,

r~’, ~9 C’)-space :

each

partially semi-algebraic

^with

respect

^{to the}

(v, s, ~, r~’, ~, I’).space ;

^the-

refore, by § 1, V, h

^is

partially semi-algebraic

^with

respect

^{to the}

(v, s)-space.

A

locally

^finite

simplicial complex (in M)

^{is a}

^locally

^{finite collection}

7f of

disjoint

^open

simplexes (1g)

such that each face of any

simplex

^{of K}

belongs

^{to K. We}

put locally

finite cellular

complex (in M)

^{is a}

locally

finite collection L of

disjoint

open cells

(17)

such that for is a finite union of cells of L.

Using

^the

regular (barycentric)

subdivision

(18)

^we

get

LEMMA 15. For any

locally

^{finite cellular}

complex

^.L

(in M)

^{there is a}

locally

^finite

simplicial complex

^g

(in M )

such that every cell of L is ^a

(finite)

^{union of}

simplexes

^{of K.}

§

^3.

Triangulation ^theorem.

Let M be a real

analytic manifold, B

^{a subset of an} affine space A.

We say that ^a map

~: j~ 2013~ M

^has

property (A)

^{iff its}

graph (in

^{A x}

M)

is

partially semi-algebraic

^with

respect

^{to A}

(19), then, by ^§ l, V, 2,

^the

image

of any

semi-algebraic

^{set is}

semi-analytic.

The

following

theorem will be

proved in §

^4.

THEOREM 1. Let

(B~)

^{be a}

locally

finite collection of

semi-analytic

^sub-

sets of a finite dimensional affine space M. There exist a

locally

finite sim-

plicial complex K with I I( I

^{= M and a}

homeomorphism

^{z : M -~. M}

(onto M)

such that:

(16) ^An open simplex ^{in M is a} subset of the form

, V ^V

with c0 ,... , ct independent; ^{its faces are the} simplexes

cYO ... eY8 with yo ^{... 7, -}

(t7) ^An open ^{cell is a convex} open bounded ^subset of an affine subspace ^{of M.}

’

(18) ^See

[181,

p. 358, ^or

[1],

^{p. 131-132.}

(19) ^{Thus a} bijection g :

E -~

^{F : where E, F are} subsets of affine

‘spaces,

^has property (~) ^iff

g-1

^has property

(.A- ).

(a) r

^has

property ( A ),

(b)

^{for any 0 E}

K, z (o)

^{is an}

analytic

submanifold

(of M)

^and

is an

analytic isomorphism,

^I

(c)

for any ^{0 E} If and _a,ny

Bv

^{we have T}

(Q) ^{C By}

^{or T}

(a)

^c

M’~ By . By

^Grauert

imbedding

^theorem

[3]

^{which ensures} that every countable real

analytic

manifold is

isomorphic

^{with a closed}

analytic

submanifold of

a

(real)

^affine space, it follows

easily

THEOREM 2. Let be ^a

locally

finite collection of

semi-analytic

subsets of a conntable real

analytic

manifold M. There exist a

locally

^finite

simplicial complex

^.g

(in

^an affine space

A)

^{and a}

homeomorphism

^7::

IK ( -~M

(onto M)

such that the

properties (a), (b), (c)

^hold.

In the case of a finite collection of bounded

semi-algebraic

^sets

the

proof

of the theorem does not

require

^{the use of lemma 10. Therefore}

we deal with

only semi-algebraic

^sets

(see § 1, VI),

^{and we obtain.}

THEOREM 3. Let

B~ ,

^{... ,}

Bk

be bounded

semi algebraic

subsets of ^an affine space M. There exist ^a finite

simplicial complex

8 in M and a se-

’I-

mi-algebraic homeomorphism ~

^{1 such that the}

properties (b), (c)

hold.

’

From this we deduce

(see § 1, VI) :

THEOREM 4. Let _{... ,}

Bk

^be

semi-algebraic

subsets of a

multiprojec-

tive _space M. There exist ^a finite

simplicial complex K (in

^{an affine}

space)

and ^a

semi-algebraic homeomorphism « : I K I (onto M)

^{such that the}

properties (b), (c)

^hold.

In

fact,

^{it is} sufficient to observe ^{that the}

semi-algebraic

map

and , is ^an

analytic imbedding

^{of P in}

1Rn2

v

(i.

^e.

yields

^an

analytic isomorphism

^{of P with an}

analytic

submanifold of

1RIl’).

§ 4.

^Proof.

In

this §

^we will prove the theorem ^1. The affine space

(of

^the

theorem)

will be denoted

by ,,11 t,

^{7 its dimension}

^by

^{n. The}

^proof

^will

proceed by

^in-

duction with

respect to n,

the theorem

being

trivial for n == 1. Thus we

consider the case n

&#x3E;

^{1 and we assume that} the theorem is true for the affine spaces of dimension n - 1.